The bielliptic locus in genus 11

Abstract

The Chow ring of ${\mathcal{M}}_g$ is known to be generated by tautological classes for $g\le 9$. Meanwhile, the first example of a non-tautological class on ${\mathcal{M}}_g$ is the fundamental class of the bielliptic locus in ${\mathcal{M}}_{12}$, due to van Zelm. It remains open if the Chow rings of ${\mathcal{M}}_{10}$ and ${\mathcal{M}}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus $10$, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported on the bielliptic locus are tautological in genus $11$. By Looijenga's vanishing theorem, this implies that they all vanish.

14 pages, comments welcome!